Optimization Practice
1. Given a 500-foot roll of fencing and a large field, what are the dimensions of the rectangle with
maximum area that can be enclosed?
2. Prove that the rectangle of maximum area for a given perimeter will always be a square.
3. A rancher has 1200 feet of fencing to enclose two adjacent rectangular corrals of equal lengths and
widths. What is the maximum area that can be enclosed in the fencing?
4. Your factory produces lemon-scented widgets. You know that the more you produce, the cheaper the
widgets. You also know that costs will eventually go up if you make too many widgets, due to the
costs of storage of the overstock. The guy in accounting says that your cost for producing x thousands
of units a day can be approximated by the formula C ( x) = 0.04 x 2 − 8.50 x + 25302 . Find the daily
production level that will minimize your costs.
5. If a car dealership sets the price of their cars at $28000, they will sell 54 cars. Every time they drop
the price by $1000, 2 more cars will be sold. What price should they set to maximize sales?
6. You run a canoe-rental business on a small river in Ohio. You currently charge $12 per canoe and
average 36 rentals a day. An industry journal says that for every fifty-cent increase in rental price, the
average business can expect to lose two rentals a day. What should you charge to maximize your
income?
7. An object is launched at 19.6 m/s from a 58.8 m tall platform. The equation for the object’s height s at
time t seconds after launch is s(t ) = −4.9t 2 + 19.6t + 58.8 . When does the object reach its maximum
height?
8. Kai realizes that his test grade is a function of the number of hours studied and knows from past
experience that 1 hour of studying will result in a grade of 66%, 2 hours gets him a grade of 75%, and
5 hours earns him 90%.
a. If Kai’s grade is a quadratic function f of the number of hours studied (x) determine an equation
for f ( x ) .
b. What is the minimum amount of time Kai must study to get a grade of at least 80%?
c. What is the maximum grade that Kai can get on the exam?